TEN SIMPLE RULES FOR MATHEMATICAL WRITING - M.I.T. APRIL 2002 Dimitri Bertsekas
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1
TEN SIMPLE RULES
FOR
MATHEMATICAL WRITING
Dimitri Bertsekas
M.I.T.
APRIL 2002
Ten Simple Rules, D. P. Bertsekas2
ON WRITING
• “Easy reading is damn hard writing”
(Hawthorne)
• “Word-smithing is a much greater percentage
of what I am supposed to be doing in life than I
would ever have thought” (Knuth)
• “I think I can tell someone how to write but I
can’t think who would want to listen” (Halmos)
Ten Simple Rules, D. P. Bertsekas3
WHAT IS MATH WRITING?
• Writing where mathematics is used as a
primary means for expression, deduction, or
problem solving.
• Examples that are:
– Math papers and textbooks
– Analysis of mathematical models in engineering, physics,
economics, finance, etc
• Examples that are not:
– Novels, essays, letters, etc
– Experimental/nonmathematical papers and reports
Ten Simple Rules, D. P. Bertsekas4
WHAT IS DIFFERENT ABOUT
MATH WRITING?
• Math writing blends two languages (natural
and math)
– Natural language is rich and allows for ambiguity
– Math language is concise and must be unambiguous
• Math writing requires slow reading
– Often expresses complex ideas
– Often must be read and pondered several times
– Often is used as reference
– Usually must be read selectively and in pieces
Ten Simple Rules, D. P. Bertsekas5
WHY THIS TALK?
• Experience is something you get only after you need it
…
• One current model: The conversational style
– “Mathematics should be written so that it reads like a
conversation between two mathematicians on a walk in the
woods” (Halmos)
– “Talk to your readers as you write” (Strang)
– Very hard to teach to others (“Effective exposition is not a
teachable art. There is no useful recipe …” Halmos)
– Controversial (where do proofs start and end? … I am not sure
what the assumptions are … I can’t find what I need … etc)
• Instead we will advocate a structured style
– Offers specific verifiable rules that students can follow and
thesis advisors can check
– Allows room to develop and improve over time
Ten Simple Rules, D. P. Bertsekas6
SOURCES
• General style books
– Strunk and White, “The Elements of Style” (www)
– Fowler and Aaron, “The Little Brown Handbook”
– Venolia, “Write Right!”
• Halmos, “How to Write Mathematics”
• Knuth, et al, “Mathematical Writing” (www)
• Kleiman, “Writing a Math Phase Two Paper,” MIT (www)
• Krantz, “A Primer of Mathematical Writing”
• Higham, “Handbook of Writing for the Mathematical
Sciences”
• Alley, “The Craft of Scientific Writing”
• Thomson, “A Guide for the Young Economist”
Ten Simple Rules, D. P. Bertsekas7
RULES OF THE GAME
• Small rules:
– Apply to a single sentence (e.g., sentence structure rules,
mathspeak rules, comma rules, etc)
• Broad rules:
– Apply to the entire document
– General style and writing strategy rules
– Are non-verifiable (e.g., organize, be clear and concise,
etc)
• Composition rules (our focus in this talk):
– Relate to how parts of the document connect
– Apply to multiple sentences
– Are verifiable
Ten Simple Rules, D. P. Bertsekas8
EXAMPLES OF SMALL RULES I
• Break up long sentences into simple ones
• Mathspeak should be “readable”
– BAD: Let k>0 be an integer.
– GOOD: Let k be a positive integer or Consider an integer
k>0.
– BAD: Let x ∈ Rn be a vector.
– GOOD: Let x be a vector in Rn or Consider a vector x ∈ Rn.
• Don’t start a sentence with mathspeak
– BAD: Proposition: f is continuous.
– GOOD: Proposition: The function f is continuous.
Ten Simple Rules, D. P. Bertsekas9
EXAMPLES OF SMALL RULES II
• Use active voice (“we” is better than “one”)
• Minimize “strange” symbols within text
• Make proper use of “very,” “trivial,” “easy,”
“nice,” “fundamental,” etc
• Use abbreviations correctly (e.g., cf., i.e., etc.)
• Comma rules
• “Which” and “that” rules
• … ETC
Ten Simple Rules, D. P. Bertsekas10
EXAMPLES OF BROAD RULES
• Language rules/goals to strive for:
precision, clarity, familiarity, forthrightness,
conciseness, fluidity, rhythm
• Organizational rules (how to structure your
work, how to edit, rewrite, proofread, etc)
• “Down with the irrelevant and the trivial”
(Halmos)
• “Honesty is the best policy” (Halmos)
• “Defend your style” (against copyeditors -
Halmos)
• … ETC
Ten Simple Rules, D. P. Bertsekas11
THE TEN COMPOSITION RULES
• Structure rules (break it into digestible pieces)
– Organize in segments
– Write segments linearly
– Consider a hierarchical development
• Consistency rules (be boring creatively)
– Use consistent notation and nomenclature
– State results consistently
– Don’t underexplain - don’t overexplain
• Readability (make it easy for the reader)
– Tell them what you’ll tell them
– Use suggestive references
– Consider examples and counterexamples
– Use visualization when possible
Ten Simple Rules, D. P. Bertsekas12
1. ORGANIZE IN SEGMENTS
• “Composition is the strongest way of seeing”
(Weston)
• Extended forms of composition have a
fundamental unit:
– Novel Paragraph
– Film Scene
– Slide presentation Slide
– Evening news program News report
• Key Question: What is the fundamental unit of
composition in math documents?
• Segment: An entity intended to be read
comfortably from beginning to end
• Not too long to be tiring, not too short to lack
content and unity
Ten Simple Rules, D. P. Bertsekas13
SEGMENTATION PROCESS
• Examples of segments:
– A mathematical result and its proof
– An example
– Several related results/examples with discussion
– An appendix
– An abstract
– A conclusions section
• A segment should “stand alone” (identifiable start and
end, transition material)
• Length: 1/2 page to 2-3 pages
Ten Simple Rules, D. P. Bertsekas14
SEGMENT STRUCTURE
Title (optional)
Transition Material
Definitions, Examples
Arguments, Illustrations
Transition Material
Ten Simple Rules, D. P. Bertsekas15
EXAMPLE OF SEGMENTATION:
A SECTION ON PROB. MODELS
• Sample space - Events (1 page)
• Choosing a sample space (0.5 page)
• Sequential models (0.75 page)
• Probability laws - Axioms (1.25 page)
• Discrete models (2 pages)
• Continuous models (1 page)
• Properties of probability laws (2 pages)
• Models and reality (1.25 page)
• History of probability (1page)
See Sec. 1.2 of Bertsekas and Tsitsiklis book
Ten Simple Rules, D. P. Bertsekas16
2. WRITE SEGMENTS LINEARLY
• Question: What is a good way to order the flow
of deduction and dependency?
• General rule: Arguments should be placed
close to where they are used (minimize
thinking strain)
• Similarly, definitions, lemmas, etc, should be
placed close to where they are used
• View ordering as an optimization problem
Ten Simple Rules, D. P. Bertsekas17
EXAMPLE
Level 1
1 2
Arguments
Dependency Level 2
Graph of 3 4
Arguments
Arguments
T
1 1
2 3
3 Nonlinear Linear 2
4 4
T T
Ten Simple Rules, D. P. Bertsekas18
3. CONSIDER A HIERARCHICAL
DEVELOPMENT
• Arguments/results used repeatedly may be
placed in special segments for efficiency
Lemmas 1, 2, 3 Level 1
Hierarchy
Analysis using Analysis using Analysis using Level 2
Lemmas 1 & 2 Lemmas 2 & 3 Lemmas 3 & 1 Hierarchy
• Possibly create special segments for special
material (e.g., math background, notation, etc)
• Analogy to subroutines in computer programs
Ten Simple Rules, D. P. Bertsekas19
4. USE CONSISTENT NOTATION
• Choose a notational style and stick with it
• Examples:
– Use capitals for random variables, lower case for values
– Use subscripts for sequences, superscripts for components
• Use suggestive/mnemonic notation. Examples: S for
set, f for function, B for ball, etc
• Use simple notation. Example: Try to avoid
parenthesized indexes: x(m,n) vs xmn
• Avoid unnecessary notation:
– BAD: Let X be a compact subset of a space Y. If f is a continuous
real-valued function over X, it attains a minimum over X.
– GOOD: A continuous real-valued function attains a minimum over
a compact set.
Ten Simple Rules, D. P. Bertsekas20
5. STATE RESULTS
CONSISTENTLY
• Keep your language/format simple and
consistent (even boring)
• Keep distractions to a minimum; make the
interesting content stand out
• Use similar format in similar situations
• Bad example:
– Proposition 1: If A and B hold, then C and D hold.
– Proposition 2: C’ and D’ hold, assuming that A’ and B’ are
true.
• Good example:
– Proposition 1: If A and B hold, then C and D hold.
– Proposition 2: If A’ and B’ hold, then C’ and D’ hold.
Ten Simple Rules, D. P. Bertsekas21
6. DON’T OVEREXPLAIN -
DON’T UNDEREXPLAIN
• Choose a target audience level of
expertise/background (e.g., undergraduate, 1st
year graduate, research specialist, etc)
• Aim your math to that level; don’t go much
over or under
• Explain potentially unfamiliar material in
separate segment(s)
• Consider the use of appendixes for
background or difficult/specialized material
Ten Simple Rules, D. P. Bertsekas22
7. TELL THEM WHAT YOU’LL
TELL THEM
• Keep the reader informed about where you are
and where you are going
• Start each segment with a short introduction
and perhaps a road map
• Don’t string together seemingly aimless
statements and surprise the reader with “we
have thus proved so and so”
• Announce your intentions/results, e.g., “It
turns out that so-and-so is true. To see this,
note …”
• Tell them what you told them
Ten Simple Rules, D. P. Bertsekas23
8. USE SUGGESTIVE
REFERENCES
• Frequent numbered equation/proposition
references are a cardinal sin
• Page flipping wastes the reader’s time and
breaks concentration
• Refer to equations/results/assumptions by
content/name (in addition to number), e.g.,
Bellman’s equation, weak duality theorem, etc
• Repeat simple math expressions
• Remind the reader of unusual notation, and
earlier analysis
• Dare to be repetitive (but don’t overdo it)
Ten Simple Rules, D. P. Bertsekas24
9. CONSIDER EXAMPLES AND
COUNTEREXAMPLES
• “Even a simple example will get three-quarters
of an idea across” (Ullman)
• Examples should have some spark, i.e., aim at
something the reader may have missed
• Illustrate definitions/results with examples that
clarify the boundaries of applicability
• Use counterexamples to clarify the limitations
of the analysis, and the need for the
assumptions
Ten Simple Rules, D. P. Bertsekas25
10. USE VISUALIZATION WHEN
POSSIBLE
• “A picture is worth a thousand words”
• Keep figures simple and uncluttered
• Use substantial captions
• Captions should reinforce and augment the
text, not repeat it
• Use a figure to illustrate the main idea of a
proof/argument with no constraint of math
formality
• Prefer graphs over tables
Ten Simple Rules, D. P. Bertsekas26
THE END
“Bad thinking never produces good writing”
(Lamport)
Good writing promotes good thinking …
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